A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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Why aren't these two sets of stochastic processes equal?

I'm learning about stochastic integrals now, and I don't understand the following: If $S$ and $L$ are two classes of processes where: $S=\{f(s,\omega) |f $ is progressively measurable and ...
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16 views

Independence of the components of a multidimensional Brownian motion

Let $B = (B^1, \dots, B^n)$ be an $n$-dimensional ($n \in \{1, 2, \dots\}$) Brownian motion (i.e. $B = (B_t)_{t \geq 0} \in \Omega \rightarrow (\mathbb{R}^n)^{[0,\infty)}$ has continuous paths, $B_0 = ...
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23 views

Calculate mean and correlation of a stochastic process

I am given the Stochastic process $Y_n$, where $n \in Z$ defined by: $ Y_n = X_n - X_{n-1}$ where $X_n$ is a process with independent and identically distributed geometric variables $X_n \sim G(p)$ ...
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15 views

Question from Lawler's “Intro to Stochastic Processes”

This is Exercise 8.17 from Lawler: I think $Y_t$ should satisfy everything but I'm having a bit of difficulty showing (ii). For i) $Y_0 = X^1_{\sigma_0} = X^1_0 = 0$ ii) Stuck on this one iii) ...
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1answer
19 views

Density Function of Random Variable Related to Brownian Motion

Above is my question. I've done the first two parts, that's no problem. I'm stuck on finding the density of the rv $R = W_1 / M$. I have got as far as $$g(x,y) = \frac{\partial^2}{\partial x ...
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0answers
15 views

Consequence of random walk with positive speed on a graph

Consider a random walk $X(n)$ on a vertex-transitive graph where the random walk has positive speed, i.e., $$ \lim\limits_{n \rightarrow \infty} \frac{d(X(n), X(0))}{n}= \alpha>0$$ almost surely. ...
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26 views

Hitting time is a stopping time

Can somebody help me proving that the following hitting time is a stopping time? Let $\{X_t\}_{t\ge 0}$ be a real-valued, right-continuous process, adapted to a filtration $\mathfrak{F}$ which ...
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1answer
9 views

Interperetation of tau in marokov chains

I'd like to ask you a question about the meaning of certain equation in my exercise. This concerns Markov Chains I have: $\tau =inf\{n>=1:X_n\in\{3,5\}\}$ and I have to calculate $P(\tau=1)$ ...
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1answer
21 views

Properties of Stochastic Interval

I'm reading "Limit Thoerem for Stochastic Processes" and finding it hard to calculate the Stochastic interval.For example : In proposition 2.10,$T$ is a stopping time: If $A\in\mathcal F_0$,I need ...
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1answer
26 views

Calculate a differenciation

$$a>0,$$ $$b>0,$$ $$\sigma >0$$ $X$ is the solution of : $$dX_t=aX_t(b-X_t)\,dt+\sigma X_t \, dB_t,\quad X_{0}=1 $$ I have also shown before that $$L_t=e^{(ab-\sigma^2/2)t+\sigma B_t}$$ Now ...
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1answer
50 views

martigale convergence theorems

Let $S_n = X_{1}+\cdots + X_{n}$ be a martingale satisfying $E[X_{k}^{2}]\leq k<\infty$, for all $k$. Show that $S_{n}$ obeys the weak law of large numbers: ...
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1answer
57 views

Understanding a proof of the strong Markov property of Lévy processes

I don't understand the the last sentence of a proof of the Markov property for Lévy processes given in Jochen Wengenroth's textbook "Wahrscheinlichkeitstheorie" (de Gruyter, 2008). I will appreciate ...
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0answers
13 views

zero drift brownian motions and barriers problem [duplicate]

Given two same brownian motion with no drift and different variances: $$(dG_1/G_1)= \sigma_1dW_g $$ $$(dG_2/G_2)= \sigma_2dW_g $$ and two barriers $P_1 > P_2$ assuming that $ \sigma_1 > ...
3
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1answer
12 views

Techniques to prove FDD convergence

When examining a sequence of stochastic processes $(\textbf{X}_n)$, $n\geq1$ convergence of marginals, i.e. $\mathbf{X}_n(t)\to\mathbf{X}(t)$ (in distribution) is often not too hard to establish for ...
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1answer
28 views

Does Brownian Motion return to the origin infinitely soon? [on hold]

Consider a standard unidimensional Brownian Motion $B_t$ (Wiener process). Fact: This process returns to the origin infinite number of times with probability one. Consider a stopping time $\tau = ...
2
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2answers
49 views

$\mathrm{d}f(x,t)$ this way $d\big(\,f(t,x)\big)=\frac{\partial f}{\partial t} \,dt+\frac{\partial f}{\partial x}\,dx$?

If $dX_t=a_t \,dt$ the next procedure is correct? $$\mathrm{d}\big(\,f(t,x)\big)=\frac{\partial f}{\partial t} dt+\frac{\partial f}{\partial x}dx=\frac{\partial f}{\partial t} dt+\frac{\partial ...
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0answers
21 views

Girsanov theorem [duplicate]

I work on an exercice and I have to calculate: $$E(W_{t}^2e^{(\int_{0}^{T}\theta_{s}dW_{s}-\frac{1}{2}\int_{0}^{T}\theta_{s}^2ds)})$$ $$\theta$$ is deterministic function I don't know how to ...
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0answers
20 views

Arrival times of doubly stochastic process - Cox process

Let N(t) be a doubly stochastic process modeled with two independent processes $\tilde N(t)$, a Poisson process with rate 1 and an almost positive process $\lambda(t)$. We define $N(t)$ by $\tilde ...
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1answer
13 views

Probabilities in markov chain

I have problem with calculating the probability of Markov Chain with 3 states S = {0,1,2}. I need to calculate $P(X_1=1,X_2=1|X_0=2)$. In the answers to my workbook I am given solution: ...
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0answers
14 views

Ito Formula for Poisson Process: $d{X_t}=a_t dt +b_t dN_t$

Let $X_t$ solve the SDE $d{X_t}=a_t dt +b_t dN_t$, where $N_t$ is a Poisson Process. I want to demosntrate that in this case the Ito formula is the next one, but I dont know how to achieve it. ...
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2answers
46 views

Martingale definition

To prove that one process is Martingale, generally we prove 3 things : 1) X is adapted. 2)$$ \mathbf{E} ( \vert X_n \vert )< \infty $$ 3) $$\mathbf{E} (X_{n+1}\mid X_1,\ldots,X_n)=X_n $$ I ...
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1answer
36 views

Ito Formula (Poisson basic process)

Let, $N_t$ be a Poisson process and let $X_t$ solve the SDE $d{X_t}=a_t dt +J_t dN_t$. Then, Ito´s fórmula is: $$df(t,X_t)=(\frac{\partial f}{\partial t} + \frac{\partial f}{\partial x}a_t)dt + ...
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27 views

Strongly continuous semigroup Kolmogorov forward integral equation

Let $\{ P_t \}_{t \geq 0}$ be a SCSF($\mathcal{S}$) (strongly continuous semigroup on $\mathcal{S}$) on the space $(E,\mathcal{E})$, where $E$ is a Polish space, equipped with the ...
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0answers
19 views

Limit of correlation function using transfer-matrix method

This question is about a stochastic process theory. I really very bad in this topic. That's why I have to ask for help. I may mistranslate some terms but I'll do my best to give you right information. ...
1
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1answer
26 views

One-Dimensional Jump-Diffusion Ito’s Formula

Let, $N_t$ be a Poisson process and let $X_t$ solve the SDE $d{X_t}=a_t dt +J_t dN_t$. Then, what is the correct Ito´s fórmula: i)$df(t,X_t)=(\frac{\partial f}{\partial t} + \frac{\partial ...
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1answer
27 views

Notation: the $\sigma$-algebra $\mathcal{F}_\tau^+$

I'm reading a probability textbook on stochastic processes (Jochen Wengenroth's "Wahrscheinlichkeitstheorie", de Gruyter 2008) and the following notation: "$\mathcal{F}_\tau^+$" came up in the ...
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1answer
17 views

Aperiodicity in irreducible markov chains

I am stuck at aperiodic property of irreducible markov chain. Let us consider an irreducible markov chain. It's stated herein that for an irreducible markov chain, a single aperiodic state implies ...
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29 views

Branching Process in simple random walk

Suppose we have a simple random walk on $\mathbb{Z}$ which stars at $1$, i.e. we have iid increment $X_n$ valued in $+1,-1$ with probability $\frac{1}{2}$ each and the sum $S_n=\sum_{i=1}^{n}X_n+S_0$ ...
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26 views

Stochastic Differential Equation for Time Integral of Stochastic Process

Let $X(t)$ denote standard Brownian motion $dX(t) = a X dt + X dW(t)$ with solution $X(t) = e^{a t + W(t)}$. I want to consider the time-integrated process \begin{equation} Y(t) := \int_0^t d\tau~ ...
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1answer
30 views

positiv Martingale process

I would to like to prove that the process: $$e^{\int_{0}^{T}\theta _{s}dW_{s}-\frac{1}{2}\int_{0}^{T}\theta _{s}^2ds}$$ is a martingale which is positiv and has a mean=1 $$\theta is continuous ...
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1answer
36 views

Lemme itô and Martingale [on hold]

I want to to find values of $a$, $b$ such that the process: $$e^{W_{t}^2+at+b\int_\limits{0}^{t}W_{s}^2\,ds}$$ be a martingale Could you please help me do that Thank you
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1answer
28 views

Escape time for a not absorbing state

Let $X$ be a right-continuous Feller Dynkin process. For $r>0$ we define the $\{\mathcal{F}_t\}_t$ stopping time (which is called escape time) $$\eta_r=\inf\{t\geq 0: \|X_t -X_0\|\geq r\}$$ We have ...
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23 views

First moment inequality and time-average limits

Suppose $\{A(t)\}_{t \geq 0}$ and $\{B(t)\}_{t \geq 0}$ are two non-negative stochastic processes such that $$ \frac{1}{T} \int_{s=0}^T A(s) \, {\rm d} s \stackrel{\text{a.s.}}{\rightarrow} a \in ...
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1answer
21 views

Show intersection of two algebras are not a $\sigma$-algebra

I have the following question: $\textbf{Question}:$ Let $\mathcal{F}_1$ and $\mathcal{F}_2$ be two algebras. Is $\mathcal{F}_1 \cap \mathcal{F}_2$ a $\sigma$-algebra? I believe the answer is no. I ...
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2answers
33 views

Two-dimensional Brownian motion

Let $B_1$ and $B_2$ be two $\mathbb{R}$-valued Brownian motions with $$\langle B_1,B_2\rangle=\int_0^t\rho_s ds,$$ where $\rho$ is progressively measurable with values in $(-1,1)$. We define ...
2
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1answer
37 views

Construction of the Itō integral

We fix some filtered probability space $(\Omega,\mathfrak{F},\{\mathfrak{F}_t\}_{t\in[0,T]},\mathbb{P})$. Let, for short, $L^2$ be the space of all progressively measurable processes in ...
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9 views

Stochastic process $X_t$, $X_t^2$, $X_{t^2}$ [on hold]

Can anyone explain me the difference between such stochastic processes:$X_t$, $X_t^2$, $X_{t^2}$ $X_t$ is let's say normal. How about two others? It's something to do with time, yes?
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1answer
29 views

Measurability of an integral

Let $\{X_t\}_{t\ge 0}$ be an adapted $\mathbb{R}$-valued stochastic process on some filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge 0},\mathbb{P}\}$ such that for each ...
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1answer
23 views

Doubt concerning Stochastic continuity

I know that a stochastic process $X$ is said to be stochastically continuous if $\forall s$ $$\lim_{t\rightarrow s}\;P(|X(t)-X(s)|>a) = 0.$$. But then it is also true that stochastic continuity ...
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29 views

Square of a weakly stationary process

I have to prove that if $X_t$ is a weakly stationary process, $X_t^2$ is also. It is easy to prove the part referred to the means but I do not know how to work with covariances. Thanks!
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1answer
35 views

Is $X_t := W_t^2$ a Wiener process for a Wiener process $(W_t)_{t \geq 0}$?

I'm studying for exam and found this exercise which I don't really understand: Suppose $W_t$ is standard Wiener process. Is process $X_t=W_t^2, t\geq0$ a Wiener process? So I need to show that ...
3
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1answer
36 views

Ito isometry for bounded Ito integral

Let $(W_t)_{t \in [0, T]}$ be a Brownian motion and $T$ be a finite time. If $\int^T_0 \beta_t d W_t$ is bounded and $\{ \beta_t \}_{t \in [0,T]}$ is locally integrable, I am curious whether the ...
2
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1answer
40 views

Markov Chain Detailed Balance property

I am having a hard time to understand the concept of the detailed balance; mostly because of the intermingled notation most of the resources use; which involves constant usage of random and state ...
3
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28 views
+50

A non-trivial 2D SDE cannot have the same Joint Density as a 1D SDE

This question comes from quantitative finance but I think it's true in general outside that setting. I'm trying to make sense of the idea that if a process depends on at least two noises there ...
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1answer
36 views

What is the probability of arrive either A or B at starting point K?

There are two points which are $A$ and $B$. The distance between $A$ and $B$ is $50$ meter. One person goes to $A$ with probability $\frac{1}{6}$, he goes to $B$ with probability $\frac{3}{6}$. And he ...
2
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2answers
26 views

What is this conditional probability?

I have been doing some reading for a project on quantitive finance, and I have been seeing a lot of this kind of conditional probabilities on a "$\mathcal{F}_{t_i}$": $$\mathbb{P} ...
2
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1answer
35 views

Independence of linear combinations of Brownian motions

Let $0<s\leq t\leq u\leq v$ and $W_x$ be a Brownian motion. Show that $aW_s+bW_t$ and $\frac{1}{v}W_v-\frac{1}{u}W_u$ are independent for $a,b$ satisfying $as+bt=0$. The question seems easy but ...
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0answers
27 views

Independence of Poisson processes watched only some of the time

Let $(X_t)$ and $(Y_t)$ be independent homogeneous Poisson processes with rates $\lambda,\mu > 0$, and let $t_1, t_2, \dots$ and $t_1', t_2', \dots$ be two increasing sequences of possibly infinite ...
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1answer
15 views

how to solve for Ut stochastic question [closed]

The process given by dUt = 􀀀-rUtdt + sigmadXt; U0 = u; where r,sigma are constants how to solve this equation for Ut? Thank you
0
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1answer
15 views

What are the conditions for $E[\int_0^tf(W_s,s)dW_s]=0$?

Let $W_t$ be the standard Brownian Motion. I am interested on the conditions on $f(\cdot)$ that guarantee that the expectation of the Ito integral below is zero: ...